I had covered the correct intent-filters, however the Tumblr app requires the category OPENABLE, so instead of the filters I had I’m now using:
<intent-filter>
<action android:name="android.intent.action.PICK" />
<category android:name="android.intent.category.DEFAULT" />
<data android:mimeType="image/*" />
</intent-filter>
<intent-filter>
<action android:name="android.intent.action.GET_CONTENT" />
<category android:name="android.intent.category.DEFAULT" />
<category android:name="android.intent.category.OPENABLE" />
<data android:mimeType="image/*" />
</intent-filter>
Only adding the line:
<category android:name="android.intent.category.OPENABLE" />